B Moment of Inertia of Rectangles

AI Thread Summary
The formula for the moment of inertia of rectangles, I = (1/12)Mbh^2, can lead to confusion if the axis of rotation is not clearly defined, particularly when it is perpendicular to the base. It is suggested that using an axis pointing into or out of the page may be more relevant for practical problems involving rotations in the same plane as the rectangle. Verification of the axis used in moment of inertia calculations is crucial to ensure accurate results. Additionally, the second moment of area is significant in beam studies and bending behavior, indicating a divergence in perspectives among engineers regarding important axes. For a comprehensive understanding of a rigid body's motion, the inertia tensor is essential.
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Why is the axis of rotation used for the moment of inertia of a rectangle about its center of mass, perpendicular to one of its bases and on the same plane instead of say, an axis that goes through the plane/page
I often encounter the formula: I = (1/12)Mbh^2 when dealing with moment of inertia of rectangles and got confused when I was unable to get the same result when figuring it out with integration. It seems that the axis of rotation used is a line perpendicular to one of the bases and on the plane of the page. Wouldn't it be more useful if the axis used is one that points into or out of the page since most problems involve rotations that remain on the same plane as the rectangle?
 
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I think the lesson is not to trust anybody's statement of a moment of inertia until you've checked what axis it is using. Or, more generally, make sure that the question they're answering is the one you're asking.

Also, I believe the second moment of area through axes in the plane is important in the study of beams and the way they bend. So I suspect engineers would disagree with your characterisation of which axes are important.
 
Please, see:
http://hyperphysics.phy-astr.gsu.edu/hbase/perpx.html#ppx

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